Resummations of self-energy graphs and KAM theory G. Gentile (Roma3), G.G. (The.H.E.S.) Communications in Mathematical Physics, 227, , 2002.

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1 Resummations of self-energy graphs and KAM theory G. Gentile (Roma3), G.G. (The.H.E.S.) Communications in Mathematical Physics, 227, ,

2 Hamiltonian for a rotators system H = 1 2 (A2 + B 2 )+εf(α, β) A = (A 1,...,A r ), B = (B 1,...,B n r ) α = (a 1,...,a r ), β = (β 1,...,β n r ) where f is an even trigonometric polynomial of degree N. Special unperturbed motions ω = (ω 1,...,ω r ) ν = (ν 1,...,ν r ) Z r A = ω, B = 0, ω ν > 1 C ν τ α = α 0 + ωt, β = β 0 r = n maximal tori or KAM tori Are there motions of the same type in presence of interaction?? i.e. A = ω + H(ψ), B = K(ψ) α = ψ + h(ψ), β = β 0 + k(ψ) with ψ ψ + ωt is a solution? It must be (necessarily if h = εh (1) +ε 2 h (2) +..., k =...): (ω ψ ) 2 h(ψ) = ε α f(ψ + h(ψ), β 0 + k(ψ)) (ω ψ ) 2 k(ψ) = ε β f(ψ + h(ψ), β 0 + k(ψ)) and β 0 mustbeanextremumfortheaverageover α: f(β) = f(α, β) dα (2π) r : (ω ψ ) 2 k 1 (ψ) = β f(ψ, β 0 ) 0 = β f(β 0 ) 1

3 Maximal tori: r=n. Series of Lindstedt, Newcomb, Poincaré: Let ϑ be a tree with p nodes v 0,...,v p 1 and root r. We attach to every node v a momentum Z n with f 0. The root momentum will be a unit vector n; and we define a current flowing on a line outgoing the node v ν(v) def = w<v ν w which we suppose 0 Note that 0 < N ω > c > 0 (if ω is Diophantine ) 1 v 5 r n λ 0 ν = ν λ0 v 0 0 v 1 v 3 v 6 v 7 v 2 v 4 v 8 v 9 v 10 Fig. 1: A tree ϑ with mv 0 = 2,mv 1 = 2,mv 2 = 3,mv 3 = 2,mv 4 = 2 and k = 12, and a few decorations. Only two momentum labels and one current label are explicitly written down; the indices enumerating the lines (because they are distinct) are not marked. Arrows represnt the partial ordering of the nodes defined by the tree. v 11 Define the value of a tree ϑ with distinct (i.e. labeled) branches Val(ϑ) = 1 p! ( linesλ=(v v) ϑ counting trees up to pivot equivalence. Then h (p) ν n = )( (ω ν(v)) 2 ϑ root current= ν Val(ϑ) v ϑ f ) 2

4 Siegel Bryuno Pöschel bound We say that λ = (v v) has scale n = 0, 1, 2,... bf if Then 2 n 1 < ω ν(v) 2 n N 2 2 2n if 2 n 1 < ω ν(v) 2 n (ω ν(v)) 2 Val(ϑ) 1 p! N2p F p 0 n= 2 2nN n N n def = number of lines of scalen If ν(v) ν(w) for all v > w then N n is small N n an2 n/τ p for some a > 0. We must use 2 n/τ N 1 nodes with momentum N to reach a line v v such that ω ν(v) 2 n, i.e. ν(v) = O(2 n/τ ). To find another one of the same scale we need as many new ones: hence N n = O(p2 n/τ N). ϑ con p nodi Val(ϑ) 1 p! p!4p N 2p F p ( 0 n= = 1 p! p!4p N 2p F p (2 2aN 0 2 2naN2n/τ p ) = n= n2n/τ ) p = B p A simple self-energy graph R ν out ν in 3

5 R ν out ν in This is a self-energy subgraph if the entering line and the exiting one have the same current ν, of scale n, and all the internal lines have scale m n + 3 and the their number is < a2 n/τ, i.e. not too large, and w R ν w = 0 and all subgraph lines have different currents (i.e. no self-energy sub-subgraph! simple ). Resummations of simple self-energy graphs The contribution to the value of a tree from a self-energy subgraph R inserted on the line v v is ν out (ω ν) 2 ( λ=(w w) R 1 (ω ν) 2 M R (ν) (ω ν) 2 ν w ν w (ω ν(λ)) 2 ) ν in (ω ν) 2 Let M(ν) def = R ε R M R (ν). We can insert m = 0,1,2,... self-energy subgraphs on every line of a tree without any such subgraph m=0 = 1 ( M(ν) ) m (ω ν) ν 2 v ν (ω ν) 2 v = 1 (ω ν) 2 M(ν) that is a convergent sum because of the Siegel Bryuno Pöschel bound. 4

6 Cancellations This is not enough because (ω ν) 2 M(ν) can vanish!! Nevertheless one shows that M(ν) = (ω ν) 2 m 1 ε(ν) andthepropagatorbecomes(ω ν) 2 (1+m 1 1 def ε(ν)) = (ω ν) 2 G (1) (ν), i.e. we have eliminated the self-energy subgraphs not containing other self-energy subgraphs. Elimination of overlapping graphs Define m 2 ε(ν) in the same way : considering all trees with simple of self-energy graphs at most and define their value as in the preceding case making use, however, of the new propagators. Then iterate indefinitely: one can check that G (k) ε (ν) converges to a limit G ( ) ε (ν). The torus invariant equation is therefore obtained by considering all the graphs without self-energies and computing them by means of the new propagator (ω ν) 2 (1+m ε (ν)) 1 def = (ω ν) 2 G ( ) (ν) which, by the Siegel Bryuno Pöschel bound does not present convergence problems and in fact this yields an algorithm to evaluate the sum of the LNP series. 5

7 Lower dimensional tori (Resonances) If f(α, β) = ν,µ eiν α+iµ β f ν,µ Feynman s rules undergo some minor changes. After resummation of the self-energy subgraphs (defined in the same way) the propagator is a Hermitian matrix (n n as before) which has the form α β α (r r) (r (n r)) = β (r r) ((n r) (n r)) ( ( (ω ν) 2 (1+O(ε 2 )) i(ω ν)bε+o(ε 2 ) = i(ω ν)bε+o(ε 2 ) (ω ν) 2 ε β f(β 0 )+O(ε 2 ) ) ) 1 where the α α elements account for the cancellations discussed in the maximal cases. Also the α β terms show cancellations (of lower order: 1 instead of 2 when ω ν 0). Nevertheless the β β elements can vanish on or near the set of infinitely many points ε for which (ω ν) 2 ε β f(β 0 ) = 0. Ifε > 0 and β 0 is amaximumthere isno 0eigenvalue andthe eigenvalues are bounded from below by (ω ν) 2. Hence on falls back in the same situation met in the maximal tori case. Convergence takes place in the domain D γ (1 > γ > 0) of complex ε where (ω ν) 2 ε β f(β 0 ) γ(ω ν) 2. The domain has the form complex ε plane O Fig.3: Analyticity domain D 0 for the lower dimensional invariant tori. The cusp at the origin is a second order one. The figure refers to the hyperbolic case. 6

8 complex ε plane Fig.4: Can the domain D 0 in Fig.3 be extended? the domain might perhaps be (near the origin) as in the picture. It reaches the real axis in cusps with apex at a set Iε 0 ; in the complex ε plane they correspond to elliptic tori which would therefore be analytic continuations of the hyperbolic tori. The analytic continuation could be continuous across the real axis on Iε 0 and Iε 0 /ε 0 ε0 1 (i.e. Iε 0 is very large near 0. 7

9 Short Bibliography Low dimensional tori S.M. Graff: On the conservation for hyperbolic invariant tori for Hamiltonian systems, J. Differential Equations 15 (1974), V.K. Mel nikov: On some cases of conservation of conditionally periodic motions under a small change of the Hamiltonian function, Soviet Math. Dokl. 6(1965), ; A family of conditionally periodic solutions of a Hamiltonian systems, Soviet Math. Dokl. 9 (1968), J. Moser: Convergent series expansions for almost periodic motions, Math. Ann. 169 (1967), Eliasson, L.H.: Absolutely convergent series expansions for quasi-periodic motions, Math. Phys. Electronic J., < mpej.unige.ch>, 2 (1996), Paper 4 A. Jorba, R. Llave, M. Zou: Lindstedt series for lower dimensional tori, in Hamiltonian systems with more than two degrees of freedom (S Agary, 1995), , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. Vol. 533, Ed. C. Simy, Kluwer Academic Publishers, The bound of Siegel-Bryuno-Pöschel A.D. Bryuno: Analytic form of differential equations. The, II. (Russian), Trudy Moskov. Mat. Obšč. 25 (1971), ; ibid. 26 (1972), J. Pöschel: Invariant manifolds of complex analytic mappings near fixed points, in Critical Phenomena, Random Systems, Gauge Theories, Les Houches, Session XLIII (1984), Vol. II, , Ed. K. Osterwalder & R. Stora, North Holland, Amsterdam, Graph methods L. Chierchia, C. Falcolini: Compensations in small divisor problems, Comm. Math. Phys. 175 (1996), G. Gallavotti: Twistless KAM tori, Comm. Math. Phys. 164 nd (1994), no. 1, Twistless KAM tori, quasiflat homoclinic intersections, 8

10 and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review, Rev. Math. Phys. 6 (1994), no. 3, More recent papers J. Bricmont, A. Kupiainen, A. Schenkel: Renormalization group for the Melnikov problem for PDE s, Comm. Math. Phys. 221 (2001), no. 1, J. Xu, J. You: Persistence of Lower Dimensional Tori Under the First Melnikov s Non-resonance Condition, Nanjing University Preprint, 1 21, 2001, J. Math. Pures Appl., in press. Xiaoping Yuan: Construction of quasi-periodic breathers via KAM techniques Comm. Math. Phys., in press. 9